Let \(\mathcal {T}_{n}\)  be the set of diffeomorphism classes of closed,
oriented and smooth manifolds of dimension \(n\) . A free circle action \(S^{1}\times M\rightarrow M\)  on a manifold \(M\in \mathcal {T}_{n}\)  is called
regular if the quotient space \(M/S^{1}\) , with the induced topology
and orientation, belongs to \(\mathcal {T}_{n-1}\) . For a given \(M\in \mathcal {T}_{n}\)  we would like to known whether there is a regular circle action on \(M\) . If the answer is affirmative, we would like to decide furthermore the
diffeomorphism types of the quotients \(M/S^{1}\)  resulted by such actions. If
\(M\)  is a homotopy sphere, these problems have been studied by
Montgemory-Yang, Edmonds, Schultz, Madsen-Milgram and Wall [1]}, [2]}, [3]}, [4]}, [5]}. In the case \(M\)  is an 1-connected 5-manifolds, the
problems have been solved by Duan and Liang [6]}.
